[11].. [12 13] (anti-windup control AWC) [14]. AWC.. L 2. 2 (6 DOF model of SFF) 2.1 (Relative position dynamic) : C ECI C L 1 [15]. (1)m f /m

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1 : (2011) Control Theory & Applications Vol. 28 No. 3 Mar ( ) L 2. ; L 2. MATLAB/Simulink L 2. : ; ; L 2; V412.4 A Adaptive synchronized control with 6 degrees of freedom and bounded input for satellite formation flight L.. U Yue-yong HU Qing-lei MA Guang-fu ZHOU Jia-kang (Department of Control Science and Engineering Harbin Institute of Technology Harbin Heilongjiang China) Abstract: An adaptive synchronized control scheme with 6 degrees of freedom(dof) and bounded input is proposed for a satellite formation flight(sff) to attenuate its disturbances. For the 6 DOF synchronized dynamic model of SFF with parameter uncertainties we first design an adaptive synchronized controller in which the external disturbance is temporary ignored. Considering the existence of disturbance we improve the controller to make the closed-loop system attenuates the disturbance with L-two-gain. Numerical simulations have been performed on the platform of MATLAB/Simulink. Results indicate that the proposed controller estimates the parameter uncertainties adaptively and attenuates the disturbance with L-two-gain while keeping a good performance in the transient process. Key words: 6 DOF synchronized control; adaptive; L-two-gain disturbance attenuation; bounded inputs 1 (Introduction) (satellite formation flying SFF). [1]. [2]. 6 (degree of freedom DOF) (3) (3).. Clohessy-Wiltshire (C-W) [3] C-W. [4] C-W. [5] [6]. [7 8]. [9]. [10] (cross-coupling) : ; : : ( ); ( ); (LC08C01); (2010RFLXG001).

2 [11].. [12 13] (anti-windup control AWC) [14]. AWC.. L 2. 2 (6 DOF model of SFF) 2.1 (Relative position dynamic) : C ECI C L 1 [15]. (1)m f /m l (2) R + ρ m f ρ + m f µ( R + ρ R 3 R ) + d 3 t = u t. (3) 1 u l (3) u t u f. ρ [16] ρ = (ẍ 2ω 0 (ẏ) ω 2 0x) i l + (ÿ 2ω 0 (ẋ) ω 2 0y) j l + z k l. (4) (4) (3) m f ρ+m f C(ω 0 ) ρ+m f N(ρ ω 0 R)+d t = u t (5) Coriolis C( ) C(ω 0 ) = 2ω (6) N( ) x µ R + ρ 3 ω2 0x N t = µ( y + R R + ρ 1 3 R ) ω2 0y. (7) µz R + ρ (Attitude dynamic) 1 Fig. 1 Schematic diagram of two-agent satellite formation C L o l x l o l y l R o l z l o l x l o l y l. R C L R = [0 0 R] T ; ρ = x i l + y j l + z k l C L. µm l m l R + R R + d 3 l = u l (1) m f ( R+ ρ)+ µm f R + ρ (R+ρ) + d 3 f = u f. (2) µ m l m f d l d f u l u f. [5] J ω + ω Jω + d a = u a (8) q 0 = 1 2 qt vω q v = T (q)ω = 1 2 [q 0I q v ]ω (9) J ω ; q = [q 0 q v ] T R 4 q 0 q v. a = [a 1 a 2 a 3 ] T a 0 a 3 a 2 a = a 3 0 a 1. (10) a 2 a 1 0 (8) : d a T GT T GT = 3µ ˆR F J ˆR F R + ρ 3 (11)

3 3 : 323 ˆR F.. F = T 1 (9)JF JF q v = JF ( T ω + T ω) = JF T ω + J ω = JF T ω ω Jω d + u. (12) (12)F T F T u = F T ((F q v ) J JF T )F q v + F T JF q v + F T d. (13) J = F T JF C = F T ((F q v ) J JF T )F u = F T u d a = F T u. (13) (5) : J q v + C q v + d a = u a. (14) 2.3 (6 DOF synchronized motion dynamic).. x = [ρ q v ] T (15) Mẍ + Cẋ + G + d = u (16) [ ] [ ] m f I C(ω 0 ) M = C = J C [ ] [ ] [ ] N t d t u t G = d = u =. 0 d a u a. 1 M : m 1 ξ 2 ξ T Mξ m 2 ξ 2 ξ R n (17) m 1 m 2. 2 M Corilis C ξ T ( 1 2Ṁ C)ξ = 0 ξ Rn. (18) 3 (Design of controller) L 2. e = x d x (19) x d = [ρ T d qvd] T T. x d L (Preliminaries) tanh ξcosh ξ tanh ξ = [tanh ξ 1 tanh ξ n ] T (20) cosh ξ = diag{cosh ξ 1 cosh ξ n } (21) ξ = [ξ 1 ξ n ] T R n. (20) (21) : d ξ tanh ξ dt tanh ξ = cosh 2 (ξ)ξ. (22) L 2 L 2 γ L 2 d L 2 [17] V (x) V (x 0 ) γ 2 t 0 d 2 dt t 0 y 2 dt (23) V (x) Lyapunov y. d y L 2 γ. 3.2 (Design of adaptive synchronized controller with bounded input).. r: r = ė + αtanh e (24) α R 6 6

4 tanh( ) (20) ṙ = ë + α cosh 2 (e)ė. (25) u = W d ˆθ + Kp tanh e + K v tanh r (34) ˆθ = P(Γ W T d r) (35) θ = [θ t θ T a ] T (26) θ a = [J 11 J 12 J 13 J 22 J 23 J 33 ] T θ t = m f. (16) [ ] [ ] W t 0 θ t Mẍ+Cẋ+G=W θ = (27) 0 W a θ a W t = ξ x 2ω 0 ẏ ω 2 µx 0x+ (x 2 +(R+y) 2 +z 2 ) 3/2 ξ y +2ω 0 ẋ ω0y+ 2 µ(r+y) (x 2 +(R+y) 2 +z 2 ) µ 3/2 R 2 µz ξ z + (x 2 +(R+y) 2 +z 2 ) 3/2 (28) W a =F T L(F ξ)+f T (F ξ) L(F ξ) F T L(F T F ξ) (29) L( ) : R 3 R 3 6 ξ 1 ξ 2 ξ ξ 1 L(ξ)= 0 ξ 1 0 ξ 2 ξ 3 0 ξ = ξ 2. (30) 0 0 ξ 1 0 ξ 2 ξ 3 (25) M (16) (24) Mṙ = Më + Mαcosh 2 (e)ė = M(ẍ d + αcosh 2 (e)ė) + C(ẋ d + αtanh e) + G u Cr. (31) (27) (31) M(ẍ d + αcosh 2 (e)ė) + C(ẋ d + αtanh e)+g u Cr = W ((ẍ d + αcosh 2 (e)ė) (ẍ d + αcosh e))θ u Cr. (32) θ θ = θ ˆθ. (33) ξ 3 W d = diag{w td W ad } K p K v Γ ; P( ) [18] : θ i θ i θ i i = (36) θ i θ i θ i. 2 x d W d ; tanh( ) ˆθ (34) u W d θ i + K p + K v. (37) K p K v α u. 1 (16) (34) (35) : λ min (K p α) > 0 (38) r λ min (K v ) ζ 1 > 0 tanh 2 ( r ) (39) r λ min (K p α)(λ min (K v ) ζ 1 tanh 2 ( r ) ) > ζ0 2 e r 2 4 tanh 2 ( e 2 )tanh 2 ( r ) (40). Lyapunov V = 1 2 rt Mr + n K pi ln(cosh e i ) θ T Γ 1 θ. (22) 1 2 λ 1tanh 2 ( y ) (41) λ 1 ln(cosh y ) V λ 2 z 2 (42) y = [e T r T ] T z = [e T r T θt ] T. Lyapunov. (41) V = 1 2 rt Ṁr + r T Mṙ + n K pi tanh e i ė i θ T Γ 1 ˆθ = r T ( 1 2Ṁ C)r + rt W θ r T W d ˆθ r T K v tanh r

5 3 : 325 tanh T (e)k p αtanh e θ T Γ 1 ˆθ = r T K v tanh r tanh T (e)k p αtanh e+ r T (W W d )θ + θ T (W T d r Γ 1 ˆθ). (43) (35) ˆθ (43) θ T (W T d r Γ 1 ˆθ) 0. [18] W W d θ ζ 0 e + ζ 1 r (44) ζ 0 ζ 1. r θ ζ 0 e r + ζ 1 r 2 = e r ζ 0 tanh e tanh r + tanh e tanh r ζ 1 r 2 tanh 2 ( r ). (45) η = [tanh e tanh r ] T (46) Q = ζ 0 2 λ min (K pα) ζ 0 e r 2 tanh e tanh r e r tanh e tanh r λ r 2 min(k v) ζ 1 tanh 2 ( r ) λ min ( ). (45)(43) (47) V η T Qη. (48) 1 (38) (40) Q L 2 (Design of adaptive L 2 -gain disturbance attenuation synchronized controller with bounded input) (34) (35). (34) L 2 u = W dˆθ + K p tanh e + K v tanh r + K r diag{tanh e i }sgn r (49) K r. 2 (16) (49) γ > 0: e 2 λ min (K p α) > 0 tanh 2 (50) ( e ) λ min (K v ) ( γ ) r 2 > 0 2 tanh 2 (51) ( r ) (ζ 0 e r λ min (K r ) 6 (tanh e i r i )) 2 < 4 tanh 2 ( e ) tanh 2 ( r ) (ζ γ λ min (K p α)(λ min (K v ) 2 ) r 2 ) tanh 2 ( r ) (52) d 0 d y L 2 γ. (41) Lyapunov. (41) (49) V = 1 2 rt Ṁr + r T Mṙ θ T Γ 1 ˆθ + n K pi tanh e i ė i = r T ( 1 2Ṁ C)r + rt (W θ W d ˆθ) r T K v tanh r K r diag{tanh e i }sgn r + r T d tanh T (e)k p αtanh e θ T Γ 1 ˆθ λ min (K p α) e 2 λ min (K v ) r 2 + ζ 0 e r + ζ 1 r 2 + r d λ min (K r ) 6 (tanh e i r i ). (53) H = V + y 2 γ 2 d 2 (54) y = [e T r T ] T. (53) (54) H = V + y 2 γ 2 d 2 λ min (K p α) e 2 λ min (K v ) r 2 + ζ 1 r 2 + ζ 0 e r + r d + y 2 γ 2 d 2 λ min (K r ) 6 (tanh e i r i ) = λ min (K v )tanh 2 ( r ) λ min (K p α)tanh 2 ( e ) + (ζ γ 2 ) r 2 ( 1 2γ r γ d ) 2 + ζ 0 e r λ min (K r ) 6 (tanh e i r i ). (55)

6 (55) η = [tanh e tanh r ] H η T Qη ( 1 2γ r γ d )2 (56) Q 11 = λ min (K p α) [ Q11 Q12 Q = Q 21 Q22 ] (57) ζ 0 e r λ min (K r ) 6 (tanh e i r i ) Q 12 = 2 tanh e tanh r ζ 0 e r λ min (K r ) 6 (tanh e i r i ) Q 21 = 2 tanh e tanh r Q 22 =λ min (K v ) (ζ γ 2 ) r 2 /tanh 2 ( r ). (50) (52) Q d 0. (56) (23) d y L 2 γ. 3 K p K v α (50) (52): 4 lim ξ 0 ξ 2 tanh 2 = 1. (58) ( ξ ) (49) u W d θ i + K p + K v + K r (59) K p α K v K r u. 4 (Simulation and analysis) MATLAB/Simulink km 0.4 m 0.4 m 0.6 m m f = 10.9 kg [9]. 100 m ρ 0 = [ ] T q 0 = [ ] T ρ d = [100 sin(ω d t) 100 cos(ω d t) 0] T 1 N m 10 N (50) (52) K p = diag{ } K v = diag{ } K r = diag{ } α = diag{ } Γ = diag{ } Fig. 2 Quaternion of the follower q d = [ ] T. 3 Fig. 3 Estimation of parameters

7 3 : Fig. 4 Comparison of attitude control torque. sin πt d t = 1 10 sin( πt π 6 ) sin( πt 10 + π 3 ) sin πt d a = 1 10 sin( πt π 6 ). sin( πt 10 + π 3 ) 4 5 ( ) ζ ζ ; ( ) ζd ; ( ) ζd + d ζ. ζ φ θ ψ (Conclusion) L 2. ; L 2. 5 Fig. 5 Comparison of relative position control force (References): [1] SCHARF D P HADAEGH F Y PLOEN S R. A survey of spacecraft formation flying guidance and control (part ): control[c] //Proceeding of the 2004 American Control Conference. New York: IEEE 2004: [2] KRISTIANSEN R NICKLASSON P J. Spacecraft formation flying: a review and new results on state feedback control[j]. Acta Astronautica (11/12): [3] CLOHESSY W H WILTSHIRE R S. Terminal guidance system for satellite rendezvous[j]. Aerospace Science (9): [4] XU Y J FITZ-COY N G. Generalized relative dynamics and control in formation flying system[c] //The 26th Annual AAS Guidance and

8 Control Conference. [S.l.]: AAS [5] AHMED J VINCENT T C DENNIS S B. Adaptive asymptotic tracking of spacecraft attitude motion with inertia matrix identification[j]. Journal of Guidance Control and Dynamics (5): [6]. [J] (3): (GAO Youtao LU Yuping XU Bo. Parameter-estimation-based control for relative attitude of spacecraft formation flying[j]. Control Theory & Applications (3): ) [7] TERUI F. Position and attitude control for a spacecraft by sliding mode control[c] //Proceedings of the 1998 American Control Conference. New York: IEEE 1998: [8] STANSBERY D T CLOUTIER J R. Position and attitude control for a spacecraft using the state-dependent riccati equation technique[c] //Proceedings of the 2000 American Control Conference. New York: IEEE 2000: [9] BAE J KIM Y PARK C. Spacecraft formation flying control using sliding mode and neural networks controller[c] //AIAA Guidance Navigation and Control Conference. [S.l.]: AIAA [10] SHAN J J. Six-degree-of-freedom synchronized adaptive learning control for spacecraft formation flying[j]. IET Control Theory Application (10): [11] SHAN J J. Synchronized attitude and translational motion control for spacecraft formation flying[j]. Journal of Aerospace Engineering (6): [12] FARRELL J POLYCARPOU M SHARMA M. On-line approximation based control of uncertain nonlinear systems with magnitude rate and bandwidth constraints on the states and actuators[c] //Proceeding of the 2004 American Control Conference. New York: IEEE 2004: [13] FARRELL J SHARMA M POLYCARPOU M. Backstepping-based flight control with adaptive function approximation[j]. Journal of Guidance Control and Dynamics (6): [14] BANG H TANK M J CHOI H D. Large angle attitude control of spacecraft with actuator saturation[j]. Control Engineering Practice (9): [15] DE QUEIROZ M S KAPILA V YAN Q G. Adaptive nonlinear control of multiple spacecraft formation flying[j]. Journal of Guidance Control and Dynamics (3): [16] KAPILA V SPARKS A G BUFFINGTON J M et al. Spacecraft formation flying: dynamics and control[c] //Proceedings of the 1999 American Control Conference. Piscataway NJ: IEEE 1999: [17] VAN DER SCHAFT A J. L 2-gain and Passivity Techniques in Nonlinear Control[M]. London: Springer-Verlag [18] DIXON W E DE QUEIROZ M S M S ZHANG F et al. Tracking control of robot manipulators with bounded torque inputs[j]. Robotica (2): : (1983 ) lyy0206@163.com; (1979 ) huqinglei@hit.edu.cn; (1962 ) magf@hit.edu.cn; (1982 ) zhou jia kang@126.com.

48 30,, []. [],,,..,,,. (The preliminaries and model of spacecraft). (Definitions and lemmas of finite-time stabilization), [3 5]. : ẋ = f(x, u), () :

48 30,, []. [],,,..,,,. (The preliminaries and model of spacecraft). (Definitions and lemmas of finite-time stabilization), [3 5]. : ẋ = f(x, u), () : 30 4 03 4 DOI: 0.764/CTA.03.07 Control Theory & Applications Vol. 30 No. 4 Apr. 03,,, (, 5000) :,,,, ;,, ;, Lyapunov.,,,,. : ; ; ; ; : V448. : A Finite-time attitude maneuver control of spacecraft under